How_I _found _A157852 _sum.nb
​​Mission: integrate the terms of the Taylor series and find a pattern for their sum.
In[]:=
l2=Normal[Series[Exp[IPix](x^(1/x)-1),{x,Infinity,3}]];
Integrate[l2,{x,1,InfinityI}]
Out[]=
MeijerG[{{},{1,1}},{{0,0,0},{}},-π]-πMeijerG[{{},{1,1,1}},{{-1,0,0,0},{}},-π]-
2
π
MeijerG[{{},{1,1,1,1}},{{-2,0,0,0,0},{}},-π]
NIntegrate[Normal[Series[Exp[IPix](x^(1/x)-1),{x,Infinity,1}]],{x,1,InfinityI}]
Out[]=
0.0576249-0.0466908
NIntegrate[Normal[Series[Exp[IPix](x^(1/x)-1),{x,Infinity,3}]],{x,1,InfinityI}]
l2
Out[]=
πx

Log[x]
x
+
2
Log[x]
2
2
x
+
3
Log[x]
6
3
x
Integrate[l2,{x,1,InfinityI}]
Out[]=
MeijerG[{{},{1,1}},{{0,0,0},{}},-π]-πMeijerG[{{},{1,1,1}},{{-1,0,0,0},{}},-π]-
2
π
MeijerG[{{},{1,1,1,1}},{{-2,0,0,0,0},{}},-π]
The above is the pattern for the sum. What does it look like?
In[]:=
l2=Normal[Series[Exp[IPix](x^(1/x)-1),{x,Infinity,3}]];
The following are the same!
∞
∫
1
(
1/x
x
-1)exp(πx)x
∞
∫
1
(
1/x
x
-1)exp(πx)x
FullSimplify[N[NIntegrate[Exp[IPix](x^(1/x)-1),{x,1,Infinity},WorkingPrecision160,MaxRecursion300],60]-NIntegrate[Exp[IPix](x^(1/x)-1),{x,1,InfinityI},WorkingPrecision60]]
Out[]=
0.×
-61
10
+0.×
-61
10

NIntegrate[l2,{x,1,Infinity}]-NIntegrate[l2,{x,1,InfinityI}]
NIntegrate
:DoubleExponentialOscillatory returns a finite integral estimate, but the integral might be divergent.
NIntegrate
:DoubleExponentialOscillatory returns a finite integral estimate, but the integral might be divergent.
Out[]=
-1.04802×
-11
10
-1.62902×
-9
10

l2
Out[]=
πx

Log[x]
x
+
2
Log[x]
2
2
x
+
3
Log[x]
6
3
x
Integrate[l2,{x,1,Infinity}]
Out[]=
1
3840
-80
4
EulerGamma
2
π
+3
6
π
+160
3
EulerGamma
π(-4+
2
π
+π(3-2Log[π]))+20
5
π
(-3+2Log[π])+10
4
π
6+HypergeometricPFQ{1,1,1,1,1},2,2,2,2,
5
2
,3,-
2
π
4
-12Log[π]+4
2
Log[π]
+2401+16HypergeometricPFQ-
1
2
,-
1
2
,-
1
2
,
1
2
,
1
2
,
1
2
,
1
2
,-
2
π
4
+8
2
Log[π]
+40
2
EulerGamma
(48+
4
π
-48π(-1+Log[π])+6
3
π
(-3+2Log[π])-6
2
π
(11-6Log[π]+2
2
Log[π]
))+640π6+6HypergeometricPFQ
1
2
,
1
2
,
3
2
,
3
2
,
3
2
,-
2
π
4
+6HypergeometricPFQ-
1
2
,-
1
2
,-
1
2
,-
1
2
,
1
2
,
1
2
,
1
2
,
1
2
,
3
2
,-
2
π
4
-9Log[π]+3
2
Log[π]
-
3
Log[π]
-2Zeta[3]+40
3
π
-49+2HypergeometricPFQ{1,1,1,1},2,2,2,2,
5
2
,-
2
π
4
+46Log[π]-18
2
Log[π]
+4
3
Log[π]
+8Zeta[3]-40
2
π
145+12HypergeometricPFQ{1,1,1},
3
2
,2,2,2,-
2
π
4
+66
2
Log[π]
-12
3
Log[π]
+2
4
Log[π]
-24Zeta[3]+2Log[π](-69+8Zeta[3])+40EulerGamma(
5
π
+96Log[π]+
4
π
(-3+2Log[π])-48π(3-2Log[π]+
2
Log[π]
)+2
3
π
(23-18Log[π]+6
2
Log[π]
)-2
2
π
(-69+66Log[π]-18
2
Log[π]
+4
3
Log[π]
+8Zeta[3]))
That is too messy!
​​​​​​
​
In[]:=
l2=Normal[Series[Exp[IPix](x^(1/x)-1),{x,Infinity,6}]];
l3=NIntegrate[l2,{x,1,InfinityI}]
Out[]=
0.070776-0.0473807
l2
Out[]=
πx

Log[x]
x
+
2
Log[x]
2
2
x
+
3
Log[x]
6
3
x
+
4
Log[x]
24
4
x
+
5
Log[x]
120
5
x
+
6
Log[x]
720
6
x
Integrate[l2,{x,1,InfinityI}]
Out[]=
MeijerG[{{},{1,1}},{{0,0,0},{}},-π]-πMeijerG[{{},{1,1,1}},{{-1,0,0,0},{}},-π]-
2
π
MeijerG[{{},{1,1,1,1}},{{-2,0,0,0,0},{}},-π]+
3
π
MeijerG[{{},{1,1,1,1,1}},{{-3,0,0,0,0,0},{}},-π]+
4
π
MeijerG[{{},{1,1,1,1,1,1}},{{-4,0,0,0,0,0,0},{}},-π]-
5
π
MeijerG[{{},{1,1,1,1,1,1,1}},{{-5,0,0,0,0,0,0,0},{}},-π]
Above is more of the pattern.
Normal[Series[Exp[IPix](x^(1/x)-1),{x,Infinity,6}]]
Out[]=
πx

Log[x]
x
+
2
Log[x]
2
2
x
+
3
Log[x]
6
3
x
+
4
Log[x]
24
4
x
+
5
Log[x]
120
5
x
+
6
Log[x]
720
6
x
Normal[Series[Exp[IPix](Log[x]/x-1),{x,Infinity,6}]]
Out[]=
πx

-1+
Log[x]
x
l2=Normal[Series[Exp[IPix](x^(1/x)-1)/x!,{x,Infinity,3}]]
Out[]=
x(-+π+Log[x])

Log[x]
2π
3/2
x
+
-Log[x]+6
2
Log[x]
12
2π
5/2
x
+
Log[x]-12
2
Log[x]
+48
3
Log[x]
288
2π
7/2
x
NIntegrate[l2,{x,1,InfinityI},WorkingPrecision100]
NIntegrate
:NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in x near {x} = {1.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000+84.33333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333}. NIntegrate obtained 0.05097929689537867126694461057376514891503278706430720899761197636759362876460307200788838682992586137-0.06819648032766032490867528421242372074940951421890844643168617781234681286045925787027565570763854972 and 2.026164271445121912017973385050998464354559521529631580418726031968202776528267209142686243352655915×
-77
10
for the integral and error estimates.
Out[]=
0.05097929689537867126694461057376514891503278706430720899761197636759362876460307200788838682992586137-0.06819648032766032490867528421242372074940951421890844643168617781234681286045925787027565570763854972
M2-%
Out[]=
0.0197967424161501322725834112565168524507219091390558185855608117960248284997789645729434944362513768+0.0208158632573095388014658777098200421762566145257348070997251769097809452723887600797710325599294149
Integrate[l2,{x,1,Infinity}]
Out[]=
$Aborted
In[]:=
Integrate[l2,{x,1,InfinityI}]
Sum[Log[x]^n/(n!x),{n,1,Infinity}]
Out[]=
-1+x
x
l2=Normal[Series[Exp[IPix](x^(1/x)-1)/(x-1)!,{x,Infinity,3}]]
Out[]=
x(-+π+Log[x])

Log[x]
2π
x
+
-Log[x]+6
2
Log[x]
12
2π
3/2
x
+
Log[x]-12
2
Log[x]
+48
3
Log[x]
288
2π
5/2
x
The following took several hours .